Human Face Recognition 
Overview of Correlation filters  

OVERVIEW OF CORRELATION FILTERS
Correlation is a natural metric for characterizing the similarity between a reference pattern r(x,y) and a test pattern t(x,y), and not surprisingly, it has been used often in pattern recognition applications. Often, the two patterns being compared exhibit relative shifts and it makes sense to compute the crosscorrelation c(τx, τy) between the two patterns for various possible shifts τx and τy as in (1); then, it makes sense to select its maximum as a metric of the similarity between the two patterns and the location of the correlation peak as the estimated shift of one pattern with respect to the other
c(τ_{x}, τ_{y}) = ∫∫ t(x,y)r(x –τ_{x} , y τ_{y})dxdy ………………………….(1)
where the limits of integration are based on the support of t(x,y). The correlation operation in (1) can be equivalently expressed as
c(τx, τy) = ∫∫T(fx, fy)R*(fx, fy) e^{J2Л}(f_{x}τ_{x} +f_{y}τ_{y})dfxdfy
= FT ‾ 1{T(fx, fy)R*(fx, fy)}………………………….(2)
where T(fx, fy)and R(fx, fy are the 2D FTs of t(x,y) and r(x,y), respectively, with fx and fy denoting the spatial frequencies. Equation (2) can be interpreted as the test pattern t(x,y) being filtered by a filter with frequency response H(fx, fy) = R*(fx, fy) to produce the output c(τx, τy) and hence the terminology “correlation filtering” for this operation. However, unlike in simple lowpass and highpass filters, the phase of the correlation filter R*(fx, fy) is very important for pattern matching. Matched Filter: The correlation filter in (2) is known as the matched filter (MF), i.e., the filter H(fx, fy)= R*(fx, fy) is simply the complex conjugate of the 2D FT of the reference pattern rđx; yŢ. It can be shown that this is optimal for detecting a reference pattern corrupted by additive white noise. But the optimality of the MF holds only if the reference pattern and the input pattern are identical except for the additive white noise and translation. In reality, the test pattern will differ from the reference pattern in many ways, e.g., rotations, scale changes, and the MF does not perform well. An interesting aside is that MFs were implemented using coherent optical processors in 1960s and much effort has gone into making optical correlators feasible. However, the devices needed for optical correlators did not advance sufficiently fast and most of the current correlation filter implementations are digital. In FR, the test face image from a subject is bound to differ from the reference face image of the same subject due to normal variations induced by expression changes, illumination differences, pose variations, and aging. In such a case, a theoretically optimum solution is to use one MF for each possible appearance of the face image. Clearly, this is computationally impractical because of the combinatorial explosion in the number of filters when we consider all possible factors that cause face appearance to change. Synthetic Discriminant Function Filters: Towards the goal of handling pattern variability using fewer correlation filters, Hester and Casasent introduced the concept of the synthetic discriminant function (SDF) filter. This approach assumes the availability of a representative set of training images and the SDF filter is a weighted sum of MFs where the weights are chosen so that the correlation outputs corresponding to the training images would yield prespecified values at the origin. For example, the correlation values (at the origin) corresponding to the training face images of authentic subjects can be set to one, and the origin values due to the impostor training images can be set to zero. The expectation is that the resulting correlation filter would yield correlation peak values close to one for nontraining face images from the authentic class and correlation peak values close to zero for images from the impostor class.
As shown schematically in Fig. 1, FR is performed by filtering the input face image with a synthesized correlation filter and processing the resulting correlation output. The correlation output is searched for a peak, and the height of that peak (relative to the background) is used to determine whether the test face image matches the training face images or not. The location of the correlation peak indicates the shift of the test face image relative to the training images.
Minimum Average Correlation Energy Filter:
Although the original SDF filter produces prespecified correlation peak values, it only controls the output at the origin for centered training images. Since the test patterns are not necessarily centered, it is nearly impossible to know where these controlled values in the output are, unless we can control the rest of the correlation plane to take on smaller values.The minimum average correlation energy (MACE) filter aims to make the controlled values (ones at the origin for centered authentic training face images) the largest values in the output plane by making the average energy of the correlation outputs as small as possible. Correlation outputs from welldesigned MACE filters typically exhibit sharp peaks for authentic input images, making the peak detection and location relatively easy and robust.
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